\subsection{Convergence of the fixed point algorithm on large single
terminal-linkage and multiple terminal-linkage class networks}\label{scn:convergence}

The fixed point iteration produces sequences that, up
to a small tolerance, satisfy \eqref{fb} at all iterates. Ideally it also
monotonically reduces the infeasibility with respect to \eqref{mak} until
convergence. Our extensive numerical experiments indicate that this is in fact
the behavior.

\begin{figure} [h] \centering \includegraphics[scale=0.3]{./Graphics/InfeasibilityVsIteration.jpg}
  \caption{Typical infeasibility of algorithm sequence, for network with a
  single terminal-linkage class} \label{fig:typical-infeas-single} \end{figure}

\begin{figure} [h]
  \centering
\includegraphics[scale=0.3]{./Graphics/InfeasibilityVsIterationMultiple.jpg}
\caption{Typical infeasibility of algorithm sequence, for network with two terminal-linkage classes}
\label{fig:typical-infeas-multiple}
\end{figure}

Figure \ref{fig:typical-infeas-single} displays the sequence of infeasibilities
$\|YA_kv_k\|_\infty$ generated by solving for a fixed point of a single terminal-linkage
network with $50$ species and $500$ complexes where at most $10$ species
participate in each complex. Figure \ref{fig:typical-infeas-multiple} displays
the analogous sequence for a network of equal size and two terminal-linkage classes.  We
have observed this (apparently linear) convergence rate consistently over all
generated networks regardless of the number of terminal-linkage classes.


\begin{figure} [h] \centering
  \includegraphics[scale=0.3]{./Graphics/SingleNetAvgIterationsVsNetSize.jpg} \caption{Average number
  of iterations for single terminal-linkage class networks}
  \label{fig:iteration-count-simple} \end{figure}

\begin{figure} [h] \centering
  \includegraphics[scale=0.3]{./Graphics/MultipleNetAvgIterationsVsNetSize.jpg} \caption{Average
  number of iterations for networks with two terminal-linkage classes}
  \label{fig:iteration-count-multiple} \end{figure}

We have also investigated the number of iterations necesary to
converge on networks of different sizes, with one and two terminal-linkage classes.
Figures \ref{fig:iteration-count-simple} and \ref{fig:iteration-count-multiple}
display the average number of iterations necessary for convergence for network
sizes ranging from $100$ to $5000$ complexes. Notably the average number of iterations
increases less than $10\%$ as the network size grows fifty-fold. (The average
was taken over $20$ instances per network size.)
This is typical of the interior-point optimization approach used by PDCO.

Our future work entails proving theoretical results on the existence of
positive equilibria for chemical reaction networks with
multiple terminal-linkage classes. However, our comprehensive numerical
experiments seem to indicate that even for networks with more than one
terminal-linkage class, there exists at least one positive fixed point.
 
